Contraction Mapping

A contraction mapping is a function between metric spaces that reduces every distance by a common factor strictly less than one. Its importance lies in the connection between this quantitative condition and the existence of a fixed point: when a contraction maps a nonempty complete metric space into itself, repeated application converges to exactly one point that the mapping leaves unchanged. This principle provides both an existence theorem and a method for constructing solutions. (jirka.org)

Definition and basic properties

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A mapping $T:X\to Y$ is a contraction if there is a constant $q$, with $0\le q<1$, such that

The number $q$ is called a contraction constant; it need not be the smallest possible one. Thus a contraction has Lipschitz continuity with a constant below one, and consequently has uniform continuity. The fixed-point results concern self-maps $T:X\to X$, although the distance inequality also makes sense between different spaces. (jirka.org)

The uniform bound is crucial: merely decreasing each nonzero distance does not necessarily provide a single factor below one. A nonexpansive mapping, by comparison, satisfies the inequality with $q=1$. Contractions need not be injective: a constant mapping is a contraction with $q=0$. Contraction is also relative to the chosen metric, rather than a property of the underlying set alone. (kconrad.math.uconn.edu)

Banach fixed-point theorem

The Banach fixed-point theorem, also called the contraction mapping principle, states that a contraction $T:X\to X$ on a nonempty complete metric space has a unique fixed point $p$, satisfying $Tp=p$. For every starting point $x_0\in X$, the fixed-point iteration

converges to $p$. Completeness means that every Cauchy sequence has a limit inside the space. The theorem applies to general metric spaces, not only to real numbers or finite-dimensional vectors. (jirka.org)

The proof combines geometric decay with completeness. Successive iterates satisfy

Using the triangle inequality and summing a geometric series gives, for $m>n$,

Hence the iterates are Cauchy and converge to a point $p$. Continuity yields $Tp=p$. If $p$ and $r$ were both fixed points, then

which forces $p=r$. (kconrad.math.uconn.edu)

Convergence and error estimates

Contraction iteration has a geometric error bound:

A useful a priori estimate, expressed through the initial step, is

An a posteriori estimate, using the latest computed step, is

These bounds turn successive differences into certified error bounds, provided the contraction hypotheses hold. They also supply stopping criteria for a numerical algorithm without requiring knowledge of the exact fixed point. A smaller contraction constant generally gives a stronger guaranteed convergence rate; an upper bound close to one may be conservative. (people.math.ethz.ch)

Examples and verification

For an affine mapping on the real line,

the contraction constant is $|a|$, and its fixed point is

Indeed, $x_n-p=a^n(x_0-p)$, directly illustrating geometric convergence. (kconrad.math.uconn.edu)

For a differentiable real function on an interval, a uniform bound on its derivative,

implies the contraction inequality through the mean value theorem. Applying the fixed-point theorem additionally requires that the function map the chosen complete domain into itself. For example, $T(x)=\cos x$ maps $[0,1]$ into itself and has contraction constant $\sin 1<1$. Iterating cosine therefore converges to the unique solution of $x=\cos x$ in that interval. (math.tecnico.ulisboa.pt)

Applications and limitations

In functional analysis, the points being iterated can themselves be functions. An important example is the Picard–Lindelöf theorem for a differential equation

One replaces it with an integral equation and defines

On a suitable closed subset of the Banach space of continuous functions, a Lipschitz bound $L$ in the dependent variable gives contraction constant at most $Lh$, where $|t-t_0|\le h$. Choosing $h$ sufficiently small and ensuring invariance establishes local existence and uniqueness. (jirka.org)

The assumptions cannot simply be omitted. As direct examples, $T(x)=x/2$ is a contraction on the incomplete space $(0,1)$, but its only possible fixed point, zero, lies outside that space. On the complete space $\mathbb R$, $T(x)=x+1$ is nonexpansive but has no fixed point. The identity mapping has every point fixed. These examples distinguish the necessity of completeness, invariance, and a uniform factor strictly below one in the theorem’s guarantee. (jirka.org)